DOI 10.1007/s12220-012-9381-6
A Sharp Comparison Theorem for Compact Manifolds with Mean Convex Boundary
Martin Man-chun Li
Received: 27 August 2012 / Published online: 14 December 2012
© Mathematica Josephina, Inc. 2012
Abstract LetM be a compactn-dimensional Riemannian manifold with nonnega- tive Ricci curvature and mean convex boundary∂M. Assume that the mean curvature H of the boundary∂MsatisfiesH≥(n−1)k >0 for some positive constantk. In this paper, we prove that the distance functiond to the boundary ∂M is bounded from above by1k and the upper bound is achieved if and only ifMis isometric to an n-dimensional Euclidean ball of radius 1k.
Keywords Comparison theorem·Ricci curvature·Mean convex boundary· Rigidity
1 Introduction
By a classical theorem of Bonnet and Myers, if a completen-dimensional Rieman- nian manifoldM has Ricci curvature at least(n−1)k, where k >0 is a constant, then the diameter ofMis at most√π
k. Applying this result to the universal coverM,˜ we see that such manifolds must be compact and have finite fundamental group. In [2], Cheng proved the rigidity theorem that if the diameter is equal to √π
k, thenMis isometric to then-sphere with constant sectional curvaturek.
In this paper, we prove a similar result for compact manifolds with nonnegative Ricci curvature and mean convex boundary. Our main result is the following.
Theorem 1.1 Let Mn be a completen-dimensional (n≥2) Riemannian manifold with nonnegative Ricci curvature and mean convex boundary∂M. Assume the mean
Communicated by Jiaping Wang.
M.M. Li (
)Mathematics Department, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada
e-mail:[email protected]
curvatureH of∂Mwith respect to the inner unit normal satisfiesH≥(n−1)k >0 for some constantk >0. Letd denote the distance function onM. Then,
x∈Msupd(x, ∂M)≤ 1
k. (1.1)
Furthermore, if we assume that∂Mis compact, thenMis also compact and equality holds in (1.1) if and only ifMn is isometric to ann-dimensional Euclidean ball of radius1k.
Remark 1.2 For any isometric embedding of a Riemannian m-manifold N into a metric space X, Gromov [5] defined the filling radius, Fill Rad (N ⊂X), to be the infimum of those numbers >0 for which N bounds in the-neighborhood U(N )⊂X, that is, the inclusion homomorphism of them-th homology (overZor Z2) Hm(N )→Hm(U(N )) vanishes. Therefore, we can restate the conclusion of Theorem1.1as Fill Rad(∂M⊂M)≤1k, and equality holds if and only ifMis the Euclidean ball of radius 1k.
Note that under the curvature assumptions in Theorem1.1, the complete manifold Mmay be non-compact. However, if we put a stronger convexity assumption on∂M, then the boundary convexity could force ∂M to be compact and hence M would also be compact. In [6], Hamilton proved that any convex hypersurface in Rn with pinched second fundamental form is compact. We conjecture that the result can be generalized to manifolds with nonnegative Ricci curvature.
Conjecture 1.3 Let Mn be a complete Riemannian n-manifold with nonempty boundary∂M. AssumeMhas nonnegative Ricci curvature and∂Mis uniformly con- vex with respect to the inner unit normal, i.e., the second fundamental formhof∂M satisfiesh≥k >0 for some constantk. Then,Mis compact andπ1(M)is finite.
Manifolds satisfying the assumptions in Conjecture1.3have been studied by sev- eral authors. Some rigidity results were obtained in [9] and [10]. In [4], J. Escobar gave upper and lower estimates for the first nonzero Steklov eigenvalue for these manifolds with boundary. However, all these results are proved under the assumption thatM is compact. Conjecture1.3above would imply that this assumption is void and these manifolds have finite fundamental group.
2 Preliminaries
In this section, we collect some known facts which will be used in the proof of The- orem1.1. LetMbe a completen-dimensional Riemannian manifold with nonempty boundary∂M. We denote by, the metric onM as well as that induced on∂M.
Supposeγ: [0, ] →M is a geodesic inM parameterized by arc length such that γ (0)andγ () lie on∂M andγ (s)lies in the interior ofM for alls∈(0, ). As- sume thatγ meets∂Morthogonally, that is,γ(0)⊥Tγ (0)∂Mandγ()⊥Tγ ()∂M.
Hence,γ is a critical point of the length functional as a free boundary problem. We
call suchγ a free boundary geodesic. For any normal vector field V along γ, the orthogonality condition implies thatV is tangent to∂Matγ (0)andγ (), hence is an admissible variation to the free boundary problem. A direct calculation gives the second variation formula of arc length:
δsγ (V , V )=
0
V(s)2−V (s)2K
γ(s), V (s) ds +
∇V ()V (), γ()
−
∇V (0)V (0), γ(0)
, (2.1)
where∇ is the Riemannian connection onM, andK(u, v)is the sectional curvature of the plane spanned byuandvinM.
LetNbe the inner unit normal of∂Mwith respect toM. The second fundamental formhof∂Mwith respect toN is defined byh(u, v)= ∇uv, N foru, vtangent to
∂M. The mean curvatureHof∂Mwith respect toNis defined as the trace ofh, that is,H=n−1
i=1h(ei, ei)for any orthonormal basise1, . . . , en−1of the tangent bundle T ∂M. The principal curvatures of∂Mare defined to be the eigenvalues ofh. Using a Frankel-type argument as in [7], we have the following lemma.
Lemma 2.1 LetM be a compact, connected n-dimensional Riemannian manifold with nonempty boundary∂M. SupposeMhas nonnegative Ricci curvature and the mean curvatureH of∂M with respect to the inner unit normal satisfiesH≥(n− 1)k >0 for some positive constantk. Then,∂Mis connected and the map
π1(∂M)−→i∗ π1(M)
induced by inclusion is surjective, i.e.,π1(M, ∂M)=0.
Proof We follow the argument given in [7]. We show under the curvature assump- tions, any free boundary geodesic must be unstable as a free boundary solution. To see this, letγ : [0, ] →M be a free boundary geodesic. Fix an orthonormal basis e1, . . . , en−1ofTγ (0)∂M, letVi(s)be the normal vector field alongγ obtained from ei by parallel translation, and using the second variation formula (2.1), we have
n−1 i=1
δ2γ (Vi, Vi)= −
0
Ric(γ(s), γ(s)) ds−Hγ ()−Hγ (0)<0,
where Ric is the Ricci curvature ofM. Therefore, δ2γ (Vi, Vi) <0 for somei, and thereforeγis unstable.
Suppose∂Mis not connected orπ1(M, ∂M)=0. In either case, there exists a free boundary geodesicγ which minimizes length in its homotopy class inπ1(M, ∂M), hence is stable. This contradicts the fact that there are no stable free boundary
geodesics inM.
We will use the following lemma, which is a special case of Theorem 1 in [8].
Lemma 2.2 LetMbe a compactn-dimensional Riemannian manifold with nonempty boundary∂Mand nonnegative Ricci curvature. If the mean curvatureHof∂Mwith respect to the unit inner normal satisfies
H≥n−1 n
|∂M|
|M| ,
where|∂M|and|M|denote the(n−1)- andn-dimensional volume of∂MandM, respectively, thenMnis isometric to a Euclidean ball.
3 Proof of Theorem1.1
In this section, we give the proof of Theorem1.1. We first prove the upper bound in (1.1). Fix any point x in the interior ofM; there exists a geodesic γ: [0, ] →M parameterized by arc length such that=d(x, ∂M)(the existence of such geodesic follows from the completeness ofM). Note thatγ lies in the interior ofMexcept at γ (). We want to prove that≤ 1k. The first variation formula tells us thatγ()is orthogonal to∂Matγ (). Moreover, the second variation ofγfor any normal vector fieldV alongγ whereV (0)=0 is nonnegative:
δ2γ (V , V )=
0
V(s)2−V (s)2K
γ(s), V (s) ds+
∇V ()V (), γ()
≥0.
(3.1) Fix an orthonormal basise1, . . . , en−1 for Tγ ()∂M, and let Ei(s) be the parallel translate ofei alongγ. DefineVi(s)=sEi(s). Substitute into (3.1) and sum overi from 1 ton−1,
n−1 i=1
δ2γ (Vi, Vi)=
0
n−1 2 −
s l
2
Ric
γ(s), γ(s)
ds−Hγ ()≥0. (3.2)
Since Ric≥0 andH≥(n−1)k >0, (3.2) implies that n−l1≥(n−1)k. Therefore, ≤1k. Since the pointxis arbitrary, we have proved inequality (1.1).
Assume now that∂Mis compact; then (1.1) implies thatMis compact. Suppose equality holds in (1.1). By rescaling the metric of M, we can assume thatk=1.
Then we want to prove thatMnis isometric to then-dimensional Euclidean unit ball.
SinceMis compact, there exists somex0in the interior ofMsuch that
d(x0, ∂M)=1. (3.3)
The key step is to show thatM is equal to the geodesic ball of radius 1 centered atx0, denoted byB1(x0). From (3.3), it is clear thatB1(x0)is contained inM. Let ρ=d(x0,·)denote the distance function from x0. SinceM has nonnegative Ricci curvature, the Laplacian comparison theorem gives
d≤n−1
d , (3.4)
where is the Laplacian operator onM, andd=d(x,·)is the distance function in Mfrom any pointx.
LetS= {q∈∂M:ρ(q)=1}. We claim thatS=∂M. To prove the claim, it suf- fices to show thatSis an open and closed subset of∂M, since∂M is connected by Lemma2.1. Note thatSis closed by the continuity ofρ. It remains to prove thatSis open in∂M. Pick any pointq∈S; we will show thatρ≡1 in a neighborhood ofq in∂M. Ifq is not a conjugate point tox0inM, then the geodesic sphere∂B1(x0)is a smooth hypersurface nearq inM, whose mean curvature with respect to the inner unit normal is at mostn−1 by the Laplacian comparison theorem (3.4). On the other hand,∂Mhas mean curvature at leastn−1 with respect to the inner unit normal by assumption. The maximum principle for hypersurfaces in manifolds [3] implies that
∂Mand∂B1(x0)coincide in a neighborhood ofq. Hence,ρ≡1 in a neighborhood ofq. Therefore,S is open near anyq which is not a conjugate point tox0inM. If qis a conjugate point ofx0, we want to show that ρ≤0 in the barrier sense [1] in a neighborhoodq, where is the Laplacian operator on∂M. Sinceqis a minimum ofρ, we can then apply the strong maximum principle in [1] for a superharmonic function in the barrier sense to conclude thatρ≡1 nearqin∂M. To see whyρis su- perharmonic in∂M, let >0 be any small constant andpbe any point on∂Mnearq. We have to find an upper barrierρwhich isC2in a neighborhood ofpin∂M, i.e., ρ(p)=ρ(p)andρ≥ρ in a neighborhood ofp in∂M. Let γ: [0,1] →Mbe a minimizing geodesic fromx0topparameterized by arc length. Letδ >0 be a small constant to be fixed later, and define
ρδ(·)=δ+d γ (δ),·
,
which is smooth in a neighborhood ofp. Notice thatρδ(p)=ρ(p)andρδ≥ρin a neighborhood ofpby the triangle inequality. By the Laplacian comparison theorem (3.4), we have
ρδ≤ n−1
d(γ (δ),·)= n−1
ρδ−δ. (3.5)
On a neighborhood ofpin∂M, we have ρδ= ρδ+H∂ρδ
∂N −Hessρδ(N, N ), (3.6) whereN is the inner unit normal of∂Mwith respect toM,H is the mean curvature of∂Mwith respect toN, and Hessρδis the Hessian ofρδinM. Observe that
ρδ(p)=ρ(p), ∂ρδ
∂N(p)= −1 and Hessρδ(N, N )(p)=0.
Choose a neighborhoodU⊂∂Mofqsuch that for anyp∈Uandδ >0 sufficiently small, we have
ρδ≥ρ≥1, ∂ρδ
∂N ≥ −1+δ and Hessρδ(N, N )≥ −δ (3.7)
on the neighborhoodU. By assumption,H≥n−1. We see from (3.5), (3.6), and (3.7) that in the neighborhoodU aroundp,
ρδ≤n−1
1−δ −(1−δ)(n−1)+δ≤
ifδis sufficiently small. Sinceis arbitrary, this shows thatρis superharmonic near qin the barrier sense and attains a local minimum atq. Therefore,ρis constant near qby the maximum principle of [1]. This proves the claim thatS=∂M.
Now, we have shown thatM=B1(x0), the geodesic ball of radius 1 centered at x0 inM. We first note thatρ is smooth up to the boundary∂M. This is true since any q∈∂M can be joined by a minimizing geodesicγ of unit length from x0 to q. As ∂M=∂B1(x0),γ is orthogonal to ∂M at q, hence is uniquely determined by q. Therefore, q is not in the cut locus of x0. Since M has nonnegative Ricci curvature, the Laplacian comparison (3.4) forρ=d(x0,·)holds in the classical sense, that is,
ρ ρ≤n−1. (3.8)
Since|∇ρ| =1 onM,ρ≡1 and ∂ρ∂ν =1 on∂M, whereν= −N is the outer unit normal of∂M, integrating (3.8) over the whole manifold M and applying Stokes’
theorem, we get
|∂M| − |M| =
∂M
ρ∂ρ
∂ν −
M
|∇ρ|2=
M
ρ ρ≤
M
(n−1)=(n−1)|M|. This implies that
1 n
|∂M|
|M| ≤1.
Since the mean curvature of∂MsatisfiesH≥n−1, by Lemma2.2,Mis isometric to a Euclidean ball of radiusr. It is clear thatr=1 asM=B1(x0). This completes the proof of Theorem1.1.
References
1. Calabi, E.: An extension of E. Hopf’s maximum principle with an application to Riemannian geome- try. Duke Math. J. 25, 45–56 (1958). MR 0092069 (19, 1056e)
2. Cheng, S.Y.: Eigenvalue comparison theorems and its geometric applications. Math. Z. 143(3), 289–
297 (1975). MR 0378001 (51 #14170)
3. Eschenburg, J.-H.: Maximum principle for hypersurfaces. Manuscr. Math. 64(1), 55–75 (1989).
MR 994381 (90c:53134)
4. Escobar, J.F.: An isoperimetric inequality and the first Steklov eigenvalue. J. Funct. Anal. 165(1), 101–116 (1999). MR 1696453 (2000h:58056)
5. Gromov, M.: Filling Riemannian manifolds. J. Differ. Geom. 18(1), 1–147 (1983). MR 697984 (85h:53029)
6. Hamilton, R.S.: Convex hypersurfaces with pinched second fundamental form. Commun. Anal.
Geom. 2(1), 167–172 (1994). MR 1312684 (95m:53078)
7. Blaine Lawson, H. Jr.: The unknottedness of minimal embeddings. Invent. Math. 11, 183–187 (1970).
MR 0287447 (44 #4651)
8. Ros, A.: Compact hypersurfaces with constant higher order mean curvatures. Rev. Mat. Iberoam.
3(3–4), 447–453 (1987). MR 996826 (90c:53160)
9. Schroeder, V., Strake, M.: Rigidity of convex domains in manifolds with nonnegative Ricci and sec- tional curvature. Comment. Math. Helv. 64(2), 173–186 (1989). MR 997359 (90h:53042)
10. Xia, C.: Rigidity of compact manifolds with boundary and nonnegative Ricci curvature. Proc. Am.
Math. Soc. 125(6), 1801–1806 (1997). MR 1415343 (97i:53043)
